A330244 Weird numbers m (A006037) such that sigma(m)/m > sigma(k)/k for all weird numbers k < m, where sigma(m) is the sum of divisors of m (A000203).

70, 10430, 1554070, 5681270, 6365870

Offset: 1

Author

Amiram Eldar, Dec 06 2019

Keywords

• nonn
• more

Comments

Benkoski and Erdős asked whether sigma(n)/n can be arbitrarily large for weird number n. Erdős offered $25 for the solution of this question.

No more terms below 10^10.

Examples

			The abundancy indices of the terms are sigma(a(n))/a(n) = 2.0571... < 2.0709... < 2.0710... < 2.0716... < 2.0716...
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, p. 77.

Links

• Stan Benkoski, Are All Weird Numbers Even?, Problem E2308, The American Mathematical Monthly, Vol. 79, No. 7 (1972), p. 774.
• S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Mathematics of Computation, Vol. 28, No. 126 (1974), pp. 617-623, alternative link, corrigendum, ibid., Vol. 29, No. 130 (1975), p. 673.
• Paul Erdős, Problems and results on combinatorial number theory III, in: M. B. Nathanson (ed.), Number Theory Day, Proceedings of the Conference Held at Rockefeller University, New York 1976, Lecture Notes in Mathematics, Vol 626, Springer, Berlin, Heidelberg, 1977, pp. 43-72. See page 47.
• Paul Erdős, Some problems I presented or planned to present in my short talk, in: B. C. Berndt, H. G. Diamond, and A. J. Hildebrand (eds.), Analytic Number Theory, Volume 1, Proceedings of a Conference in Honor of Heini Halberstam, Progress in Mathematics, Vol. 138, Birkhäuser Boston, 1996, pp. 333-335.

Crossrefs

Cf. A000203, A006037.